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In mathematics, particularly linear algebra, an orthogonal basis for an inner product space V is a basis for V whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis.

As coordinates

Any orthogonal basis can be used to define a system of orthogonal coordinates V. Orthogonal (not necessarily orthonormal) bases are important due to their appearance from curvilinear orthogonal coordinates in Euclidean spaces, as well as in Riemannian and pseudo-Riemannian manifolds.
In functional analysis

In functional analysis, an orthogonal basis is any basis obtained from an orthonormal basis (or Hilbert basis) using multiplication by nonzero scalars.

Extensions

The concept of an orthogonal (but not of an orthonormal) basis is applicable to a vector space V (over any field) equipped with a symmetric bilinear form ⟨·,·⟩, where orthogonality of two vectors v and w means ⟨v, w⟩ = 0. For an orthogonal basis {ek} :

$$\langle\mathbf{e}_j,\mathbf{e}_k\rangle = \left\{\begin{array}{ll}q(\mathbf{e}_k) & j = k \\ 0 & j \ne k \end{array}\right.\quad,$$

where q is a quadratic form associated with ⟨·,·⟩: q(v) = ⟨v, v⟩ (in an inner product space q(v) = | v |2). Hence,

$$\langle\mathbf{v},\mathbf{w}\rangle = \sum\limits_{k} q(\mathbf{e}_k) v^k w^k\ ,$$

where vk and wk are components of v and w in {ek} .

References

Lang, Serge (2004), Algebra, Graduate Texts in Mathematics 211 (Corrected fourth printing, revised third ed.), New York: Springer-Verlag, pp. 572–585, ISBN 978-0-387-95385-4
Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete 73. Springer-Verlag. p. 6. ISBN 3-540-06009-X. Zbl 0292.10016.